J. Rákosník (ed.),
Function spaces, differential operators and nonlinear analysis.
Proceedings of the conference held in Paseky na Jizerou, September 3-9, 1995.
Mathematical Institute, Czech Academy of Sciences, and Prometheus Publishing House, Praha 1996
p. 207 - 214

Entropy numbers in weighted function spaces and applications

Dorothee Haroske

Mathematisches Institut, Fakultät für Mathematik und Informatik, Friedrich-Schiller-Universität, Leutragraben 1, 07743 Jena, Germany haroske@minet.uni-jena.de

Abstract: We study weighted function spaces of Sobolev type $H^s_p(w(x))$ (or, more generally, of type $B^s_{p,q}(w(x))$ or $F^s_{p,q}(w(x))$) where the weight function is of at most polynomial growth. In particular, we investigate continuity and compactness of embedding operators between function spaces of such type. The concept of entropy numbers (as well as of approximation numbers) has turned out to be very effective in ``measuring'' compactness. Specifying the weight function to $w(x)=(1+|x|^2)^{\alpha/2}$ one may characterize the compactness of that embedding by means of its entropy numbers $e_k$. In particular, it turns out that $$ e_k \sim k^{-\mu}\qquad\text{or}\qquad e_k \sim k^{-\nu}\,(1+\log k)^\gamma, $$ resp., where the positive exponents $\mu$, $\nu$ and $\gamma$ depend upon the underlying spaces and the weight function. Afterwards one can apply these results using Carl's inequality to estimate the distribution of eigenvalues of compact operators acting in spaces of the above type, $$ B=b_2(x)\,b(x,D)\,b_1(x), $$ where $b_1$ and $b_2$ typically belong to some weighted $L_p$ or $H^s_p$ spaces and $b(x,D)$ is in the Hörmander class $\Psi^{-\kappa}_{1,\gamma},\;\kappa>0,\; 0\leq\gamma\leq 1$. Moreover, there are also applications to the theory of ``negative'' spectra of related symmetric operators in $L_2$.

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